Factored orthogonal transformations for recursive eigendecomposition

Abstract

Factorizations of orthogonal matrices play an important role in modern digital signal processing. Here we focus on their usefulness in the field of recursive eigendecomposition methods. We concentrate on recursive algorithms which use Givens rotations to update the eigenvector matrix. The orthogonality of the estimated eigenvector matrix is known to be crucial for the numerical stability of the recursive algorithms. It is shown that this property can be enforced by decomposing the orthogonal matrix into a sequence of simple plane rotations. Then a rotation method for updating the plane rotations is developed. This method has the advantage that loss of numerical accuracy is avoided while retaining the inherent parallel structure of the algorithm. Moreover, it consists solely of rotation operations. Therefore, the new method is ideally suited for execution on parallel architectures which have dedicated rotation nodes, such as a CORDIC processor.